On the Diophantine Equation Mp^x + (Mq + 1)^y = z^2

Abstract

In this paper, we study and solve the exponential Diophantine equation of the form Mxp + (Mq + 1)y = z2 for Mersenne primes Mp and Mq and non-negative integers x, y, and z. We use elementary methods, such as the factoring method and the modular arithmetic method, to prove our research results. Several illustrations are presented, as well as cases where solutions to the Diophantine equation do not exist.